| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s − 6·13-s + 16-s + 6·17-s − 2·23-s + 25-s + 4·27-s + 10·29-s − 3·36-s − 12·39-s − 2·43-s + 2·48-s − 49-s + 12·51-s + 6·52-s + 28·53-s − 6·61-s − 64-s − 6·68-s − 4·69-s + 2·75-s + 5·81-s + 20·87-s + 2·92-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s − 1.66·13-s + 1/4·16-s + 1.45·17-s − 0.417·23-s + 1/5·25-s + 0.769·27-s + 1.85·29-s − 1/2·36-s − 1.92·39-s − 0.304·43-s + 0.288·48-s − 1/7·49-s + 1.68·51-s + 0.832·52-s + 3.84·53-s − 0.768·61-s − 1/8·64-s − 0.727·68-s − 0.481·69-s + 0.230·75-s + 5/9·81-s + 2.14·87-s + 0.208·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.376411700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.376411700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.64424565668071911071373429551, −10.48578500125122597595345153564, −10.03045588248703020413097068827, −9.566490198455084588934479518599, −9.474835345348604022768159047682, −8.717431507189508441762219730357, −8.253158637349628081886891555853, −8.179094225529055222791246943536, −7.33807092747286015892275102407, −7.26015157601889847259167581047, −6.65867049724933434258730447769, −5.89706123978545200226975534533, −5.28186697409185948060013263027, −4.96054312587536855532952426758, −4.18675780044876769409520304301, −3.92534040678059442665584062182, −2.86521533116791941437839235681, −2.86349550330828889337040673671, −1.92143899985575659444802638883, −0.881998858320457016092041279158,
0.881998858320457016092041279158, 1.92143899985575659444802638883, 2.86349550330828889337040673671, 2.86521533116791941437839235681, 3.92534040678059442665584062182, 4.18675780044876769409520304301, 4.96054312587536855532952426758, 5.28186697409185948060013263027, 5.89706123978545200226975534533, 6.65867049724933434258730447769, 7.26015157601889847259167581047, 7.33807092747286015892275102407, 8.179094225529055222791246943536, 8.253158637349628081886891555853, 8.717431507189508441762219730357, 9.474835345348604022768159047682, 9.566490198455084588934479518599, 10.03045588248703020413097068827, 10.48578500125122597595345153564, 10.64424565668071911071373429551
